Good Swinging Fun! The Mathematics of a Playground Cornerstone. By: Corey Small MA 354 Final Project Fall 2007 The Mathematics of a Playground Cornerstone. By: Corey Small MA 354 Final Project Fall 2007

Wha? Swings Make Good Pendulums! The physical Forces on a Pendulum include gravity, drag, and a driving force. The Differential Equation modeling the change in angle θ with respect to time is as follows: Change to Dimensionless Variables: And Get:

Wha? Lets Look at the Mathematica Program to be Used: Presented is the Mathematica Function which will provide all the plots. The problem has been broken down into two first order differential equations, for the benefit of graphing phase space diagrams. The initial conditions are θ0 for the angle and ω0 for angular speed. The function α is used to prevent the explicit appearance of t in the two DEs.

Wha? Small Angle Approximation Makes This DE Linear. The Sin(θ) appears as simply θ in the DE. Without Drag or a Driving Force, Angle VS time is a sinusoidal function: Without Drag or a Driving Force, the Phase Space Plot is an Ellipse. This is stable.

Wha? Small Angle Approximation Makes This DE Linear. When the Damping Force is non-zero, any oscillations will eventually die out. Awwww! With Drag, Note the Exponential Decay in Amplitude. With Drag, the Phase Space Plot Spirals in towards (0,0), zero angle and zero velocity.

Wha? Getting Closer to Reality… When you kick your legs on a swing, you supply a driving force for the oscillations! Yay! Driving Force and Damping Force Present: The Phase Space Trajectory starts out wobbly, but then reaches an attractor.

Wha? Transient Effects are COOL! The Oscillations start out messy, but the driving force eventually takes hold. Look at those wild Transient Effects that eventually give way to normal oscillation: The Swirls and loops correspond to the first few transient cycles. The stable oscillations can be seen as the ellipse in the middle.

Wha? With the Sin(θ) Term Present, Things Get Really Weird: When linearity is thrown out the window, things can become Chaotic. Chaos is when the possible outcomes are extremely sensitive to initial Conditions. Just don’t know whats going to happen next! This is for small initial conditions: This is like an attractor, sort of. The oscillations collapse onto something close to periodic, but in a chaotic fashion.

Wha? My Favorite of All: This one I found by simply tweaking the values of the damping, driving, and driving frequency. This would be the best theme Park ride in the world! It doesn’t even appear that the oscillations are about θ = 0. This situation is for large initial angles, particularly θ = π/1.001 This Phase Space Plot shows unending Chaos. If the time were not limited to a finite amount, the Phase Trajectory would plot everywhere!

The End! Any Questions?